| L(s) = 1 | + (−1.73 − 1.73i)7-s + 3.46i·11-s + (−1 − i)13-s + (1 − i)17-s + 6.92·19-s + (−1.73 + 1.73i)23-s − 4i·29-s + 3.46i·31-s + (−5 + 5i)37-s − 2·41-s + (1.73 − 1.73i)43-s + (1.73 + 1.73i)47-s − 1.00i·49-s + (−7 − 7i)53-s + 6.92·59-s + ⋯ |
| L(s) = 1 | + (−0.654 − 0.654i)7-s + 1.04i·11-s + (−0.277 − 0.277i)13-s + (0.242 − 0.242i)17-s + 1.58·19-s + (−0.361 + 0.361i)23-s − 0.742i·29-s + 0.622i·31-s + (−0.821 + 0.821i)37-s − 0.312·41-s + (0.264 − 0.264i)43-s + (0.252 + 0.252i)47-s − 0.142i·49-s + (−0.961 − 0.961i)53-s + 0.901·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.623140480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.623140480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (1.73 + 1.73i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + (1 + i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1 + i)T - 17iT^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 + (1.73 - 1.73i)T - 23iT^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + (5 - 5i)T - 37iT^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + (-1.73 + 1.73i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.73 - 1.73i)T + 47iT^{2} \) |
| 53 | \( 1 + (7 + 7i)T + 53iT^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + (-5.19 - 5.19i)T + 67iT^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (-7 - 7i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (-12.1 + 12.1i)T - 83iT^{2} \) |
| 89 | \( 1 + 8iT - 89T^{2} \) |
| 97 | \( 1 + (-7 + 7i)T - 97iT^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.520749060216176963033070559859, −7.60747629462591466962290146786, −7.17701346974444167643653588760, −6.47976300391014761415905092506, −5.42856745524698743047624218384, −4.83295756773629346027627424120, −3.78884765386839525239801075157, −3.14260951826558450475979736059, −2.00228595309067914745493579280, −0.77057843385744699762407684632,
0.71709893175084664728624092150, 2.08781156711979018576304519092, 3.12616143265236607968152943633, 3.64457100768756707626991856008, 4.88805879644487132858901770370, 5.64189578287351057105372654585, 6.19599517483342415407572699460, 7.06307892700023616535790132134, 7.84418364952545564573300211369, 8.598195907856926827542290100569
